450 Prof. Sylvester on the Motion atid Rest of Fluids. 



/{udi/ + vdx), i.e. if we please f \u -j— + '^'—, — )ds 



(where & is for clearness' sake and to avoid double limits taken 

 an element of the bounding curve) as at first sight it might ap- 

 pear to be, but is in fact equal to 



/(«-4f-^4f)'''- 



I shall demonstrate this point in the next number of the 

 Magazine. It at first caused me some trouble in conducting 

 the annexed inquiry. I shall also take occasion at some other 

 time to revert to a new species (as I believe) of partial dif- 

 ferential equations ; that is to say, where there are fewer of 

 them than of the principal variables, which may be called 

 therefore Indeterminate Partial Differential Equations. A 

 complete solution of one of these appears in the subjoined 



Investigation. 



For the sake of simplicity I take an incompressible fluid. 

 The method is nowise different for a fluid of varying density. 



Let A .a? Ay A 2 be any displacement undergone by a 

 particle at the point z^ ?/, 2 parallel to the axes s, y, z re- 

 spectively ; it is easily shown that to satisfy the condition of 

 invariability of mass we must have 



dAa? d l^y d Lz _^ 

 dx dy dz ^ '" ^ '' 



One relation between m, v^ to the velocities parallel to or, y^ z 

 is obtained immediately by putting u^t, vlt, wit for A ^, 

 A^j A 2, which gives 



du d V dw _ . . 



da? dy dz ~ ^ 



as usual. 



Again, if X Y Z be the impressed forces, and X^ Y^ Z, 

 the internal forces acting on any particle parallel to the axes, 

 we have 



^ ^r du du du dw ,^ . 



' d t dx iiy d z ^ 



,^ ^ ^y. dv d 1) dv dw ,^, . 



' dt dx dy dz 



r, ri dw dw dw dw ,, . 



from the mere geometry of the question, 



